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Dornheim T, Vorberger J, Moldabekov ZA, Böhme M. Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220217. [PMID: 37393936 DOI: 10.1098/rsta.2022.0217] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Accepted: 01/10/2023] [Indexed: 07/04/2023]
Abstract
Rigorous diagnostics of experiments with warm dense matter are notoriously difficult. A key method is X-ray Thomson scattering (XRTS), but the interpretation of XRTS measurements is usually based on theoretical models that entail various approximations. Recently, Dornheim et al. [Nat. Commun. 13, 7911 (2022)] introduced a new framework for temperature diagnostics of XRTS experiments that is based on imaginary-time correlation functions. On the one hand, switching from the frequency to the imaginary-time domain gives one direct access to a number of physical properties, which facilitates the extraction of the temperature of arbitrarily complex materials without relying on any models or approximations. On the other hand, the bulk of theoretical work in dynamic quantum many-body theory is devoted to the frequency domain, and, to the best of our knowledge, the manifestation of physics properties within the imaginary-time density-density correlation function (ITCF) remains poorly understood. In the present work, we aim to fill this gap by introducing a simple, semi-analytical model for the imaginary-time dependence of two-body correlations within the framework of imaginary-time path integrals. As a practical example, we compare our new model to extensive ab initio path integral Monte Carlo results for the ITCF of a uniform electron gas, and find excellent agreement over a broad range of wavenumbers, densities and temperatures. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Zhandos A Moldabekov
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Maximilian Böhme
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
- Technische Universität Dresden, D-01062 Dresden, Germany
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2
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Moldabekov ZA, Lokamani M, Vorberger J, Cangi A, Dornheim T. Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons. J Chem Phys 2023; 158:094105. [PMID: 36889956 DOI: 10.1063/5.0135729] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023] Open
Abstract
We assess the accuracy of common hybrid exchange-correlation (XC) functionals (PBE0, PBE0-1/3, HSE06, HSE03, and B3LYP) within the Kohn-Sham density functional theory for the harmonically perturbed electron gas at parameters relevant for the challenging conditions of the warm dense matter. Generated by laser-induced compression and heating in the laboratory, the warm dense matter is a state of matter that also occurs in white dwarfs and planetary interiors. We consider both weak and strong degrees of density inhomogeneity induced by the external field at various wavenumbers. We perform an error analysis by comparing with the exact quantum Monte Carlo results. In the case of a weak perturbation, we report the static linear density response function and the static XC kernel at a metallic density for both the degenerate ground-state limit and for partial degeneracy at the electronic Fermi temperature. Overall, we observe an improvement in the density response when the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are used, compared with the previously reported results for the PBE, PBEsol, local-density approximation, and AM05 functionals; B3LYP, on the other hand, does not perform well for the considered system. Additionally, the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are more accurate for the density response properties than SCAN in the regime of partial degeneracy.
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Affiliation(s)
- Zhandos A Moldabekov
- Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826 Görlitz, Germany
| | - Mani Lokamani
- Information Services and Computing, Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Jan Vorberger
- Insitute of Radiation Physics, Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Attila Cangi
- Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826 Görlitz, Germany
| | - Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826 Görlitz, Germany
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Moldabekov ZA, Lokamani M, Vorberger J, Cangi A, Dornheim T. Non-empirical Mixing Coefficient for Hybrid XC Functionals from Analysis of the XC Kernel. J Phys Chem Lett 2023; 14:1326-1333. [PMID: 36724891 PMCID: PMC9923747 DOI: 10.1021/acs.jpclett.2c03670] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2022] [Accepted: 01/13/2023] [Indexed: 06/18/2023]
Abstract
We present an analysis of the static exchange-correlation (XC) kernel computed from hybrid functionals with a single mixing coefficient such as PBE0 and PBE0-1/3. We break down the hybrid XC kernels into the exchange and correlation parts using the Hartree-Fock functional, the exchange-only PBE, and the correlation-only PBE. This decomposition is combined with exact data for the static XC kernel of the uniform electron gas and an Airy gas model within a subsystem functional approach. This gives us a tool for the non-empirical choice of the mixing coefficient under ambient and extreme conditions. Our analysis provides physical insights into the effect of the variation of the mixing coefficient in hybrid functionals, which is of immense practical value. The presented approach is general and can be used for other types of functionals like screened hybrids.
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Affiliation(s)
- Zhandos A. Moldabekov
- Center
for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826Görlitz, Germany
| | - Mani Lokamani
- Information
Services and Computing, Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328Dresden, Germany
| | - Jan Vorberger
- Institute
of Radiation Physics, Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328Dresden, Germany
| | - Attila Cangi
- Center
for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826Görlitz, Germany
| | - Tobias Dornheim
- Center
for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826Görlitz, Germany
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Moldabekov Z, Böhme M, Vorberger J, Blaschke D, Dornheim T. Ab Initio Static Exchange-Correlation Kernel across Jacob's Ladder without Functional Derivatives. J Chem Theory Comput 2023; 19:1286-1299. [PMID: 36724889 PMCID: PMC9979610 DOI: 10.1021/acs.jctc.2c01180] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/03/2023]
Abstract
The electronic exchange─correlation (XC) kernel constitutes a fundamental input for the estimation of a gamut of properties such as the dielectric characteristics, the thermal and electrical conductivity, or the response to an external perturbation. In this work, we present a formally exact methodology for the computation of the system specific static XC kernel exclusively within the framework of density functional theory (DFT) and without employing functional derivatives─no external input apart from the usual XC-functional is required. We compare our new results with exact quantum Monte Carlo (QMC) data for the archetypical uniform electron gas model under both ambient and warm dense matter conditions. This gives us unprecedented insights into the performance of different XC functionals, and it has important implications for the development of new functionals that are designed for the application at extreme temperatures. In addition, we obtain new DFT results for the XC kernel of warm dense hydrogen as it occurs in fusion applications and astrophysical objects. The observed excellent agreement to the QMC reference data demonstrates that presented framework is capable to capture nontrivial effects such as XC-induced isotropy breaking in the density response of hydrogen at large wave numbers.
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Affiliation(s)
- Zhandos Moldabekov
- Center
for Advanced Systems Understanding (CASUS), D-02826Görlitz, Germany,Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328Dresden, Germany,E-mail:
| | - Maximilian Böhme
- Center
for Advanced Systems Understanding (CASUS), D-02826Görlitz, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328Dresden, Germany
| | - David Blaschke
- Institute
of Theoretical Physics, University of Wroclaw, 50-204Wroclaw, Poland
| | - Tobias Dornheim
- Center
for Advanced Systems Understanding (CASUS), D-02826Görlitz, Germany,Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328Dresden, Germany,E-mail:
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Dynamical Density of Two-Dimensional Dusty Plasmas. IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY, TRANSACTIONS A: SCIENCE 2022. [DOI: 10.1007/s40995-022-01386-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Moldabekov Z, Vorberger J, Dornheim T. Density Functional Theory Perspective on the Nonlinear Response of Correlated Electrons across Temperature Regimes. J Chem Theory Comput 2022; 18:2900-2912. [PMID: 35484932 PMCID: PMC9097288 DOI: 10.1021/acs.jctc.2c00012] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2022] [Indexed: 11/28/2022]
Abstract
We explore a new formalism to study the nonlinear electronic density response based on Kohn-Sham density functional theory (KS-DFT) at partially and strongly quantum degenerate regimes. It is demonstrated that the KS-DFT calculations are able to accurately reproduce the available path integral Monte Carlo simulation results at temperatures relevant for warm dense matter research. The existing analytical results for the quadratic and cubic response functions are rigorously tested. It is demonstrated that the analytical results for the quadratic response function closely agree with the KS-DFT data. Furthermore, the performed analysis reveals that currently available analytical formulas for the cubic response function are not able to describe simulation results, neither qualitatively nor quantitatively, at small wavenumbers q < 2qF, with qF being the Fermi wavenumber. The results show that KS-DFT can be used to describe warm dense matter that is strongly perturbed by an external field with remarkable accuracy. Furthermore, it is demonstrated that KS-DFT constitutes a valuable tool to guide the development of the nonlinear response theory of correlated quantum electrons from ambient to extreme conditions. This opens up new avenues to study nonlinear effects in a gamut of different contexts at conditions that cannot be accessed with previously used path integral Monte Carlo methods.
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Affiliation(s)
- Zhandos Moldabekov
- Center
for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Tobias Dornheim
- Center
for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
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Dornheim T, Moldabekov ZA, Vorberger J, Militzer B. Path integral Monte Carlo approach to the structural properties and collective excitations of liquid [Formula: see text] without fixed nodes. Sci Rep 2022; 12:708. [PMID: 35027602 PMCID: PMC8758733 DOI: 10.1038/s41598-021-04355-9] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2021] [Accepted: 12/21/2021] [Indexed: 11/10/2022] Open
Abstract
Due to its nature as a strongly correlated quantum liquid, ultracold helium is characterized by the nontrivial interplay of different physical effects. Bosonic [Formula: see text] exhibits superfluidity and Bose-Einstein condensation. Its physical properties have been accurately determined on the basis of ab initio path integral Monte Carlo (PIMC) simulations. In contrast, the corresponding theoretical description of fermionic [Formula: see text] is severely hampered by the notorious fermion sign problem, and previous PIMC results have been derived by introducing the uncontrolled fixed-node approximation. In this work, we present extensive new PIMC simulations of normal liquid [Formula: see text] without any nodal constraints. This allows us to to unambiguously quantify the impact of Fermi statistics and to study the effects of temperature on different physical properties like the static structure factor [Formula: see text], the momentum distribution [Formula: see text], and the static density response function [Formula: see text]. In addition, the dynamic structure factor [Formula: see text] is rigorously reconstructed from imaginary-time PIMC data. From simulations of [Formula: see text], we derived the familiar phonon-maxon-roton dispersion function that is well-known for [Formula: see text] and has been reported previously for two-dimensional [Formula: see text] films (Nature 483:576-579 (2012)). The comparison of our new results for both [Formula: see text] and [Formula: see text] with neutron scattering measurements reveals an excellent agreement between theory and experiment.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), 02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany
| | - Zhandos A. Moldabekov
- Center for Advanced Systems Understanding (CASUS), 02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany
| | - Burkhard Militzer
- Department of Earth and Planetary Science, University of California, Berkeley, CA 94720 USA
- Department of Astronomy, University of California, Berkeley, CA 94720 USA
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Shahzad A, Manzoor A, Wang W, Mahboob A, Kashif M, He MG. Dynamic Characteristics of Strongly Coupled Nonideal Plasmas. ARABIAN JOURNAL FOR SCIENCE AND ENGINEERING 2021. [DOI: 10.1007/s13369-021-05954-4] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Dornheim T, Cangi A, Ramakrishna K, Böhme M, Tanaka S, Vorberger J. Effective Static Approximation: A Fast and Reliable Tool for Warm-Dense Matter Theory. PHYSICAL REVIEW LETTERS 2020; 125:235001. [PMID: 33337174 DOI: 10.1103/physrevlett.125.235001] [Citation(s) in RCA: 19] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2020] [Revised: 10/12/2020] [Accepted: 11/03/2020] [Indexed: 06/12/2023]
Abstract
We present an effective static approximation (ESA) to the local field correction (LFC) of the electron gas that enables highly accurate calculations of electronic properties like the dynamic structure factor S(q,ω), the static structure factor S(q), and the interaction energy v. The ESA combines the recent neural-net representation by T. Dornheim et al., [J. Chem. Phys. 151, 194104 (2019)JCPSA60021-960610.1063/1.5123013] of the temperature-dependent LFC in the exact static limit with a consistent large wave-number limit obtained from quantum Monte Carlo data of the on-top pair distribution function g(0). It is suited for a straightforward integration into existing codes. We demonstrate the importance of the LFC for practical applications by reevaluating the results of the recent x-ray Thomson scattering experiment on aluminum by Sperling et al. [Phys. Rev. Lett. 115, 115001 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.115001]. We find that an accurate incorporation of electronic correlations in terms of the ESA leads to a different prediction of the inelastic scattering spectrum than obtained from state-of-the-art models like the Mermin approach or linear-response time-dependent density functional theory. Furthermore, the ESA scheme is particularly relevant for the development of advanced exchange-correlation functionals in density functional theory.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
| | - Attila Cangi
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
| | - Kushal Ramakrishna
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
- Technische Universität Dresden, D-01062 Dresden, Germany
| | - Maximilian Böhme
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Technische Universität Dresden, D-01062 Dresden, Germany
| | - Shigenori Tanaka
- Graduate School of System Informatics, Kobe University, Kobe 657-8501, Japan
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
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10
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Dornheim T, Vorberger J. Finite-size effects in the reconstruction of dynamic properties from ab initio path integral Monte Carlo simulations. Phys Rev E 2020; 102:063301. [PMID: 33466040 DOI: 10.1103/physreve.102.063301] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2020] [Accepted: 11/15/2020] [Indexed: 06/12/2023]
Abstract
We systematically investigate finite-size effects in the dynamic structure factor S(q,ω) of the uniform electron gas obtained via the analytic continuation of ab initio path integral Monte Carlo data for the imaginary-time density-density correlation function F(q,τ). Using the recent scheme by Dornheim et al. [Phys. Rev. Lett. 121, 255001 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.255001], we find that the reconstructed spectra are not afflicted with any finite-size effects for as few as N=14 electrons both at warm dense matter (WDM) conditions and at the margins of the strongly correlated electron liquid regime. Our results further corroborate the high quality of our current description of the dynamic density response of correlated electrons, which is of high importance for many applications in WDM theory and beyond.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
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Dornheim T, Vorberger J, Bonitz M. Nonlinear Electronic Density Response in Warm Dense Matter. PHYSICAL REVIEW LETTERS 2020; 125:085001. [PMID: 32909774 DOI: 10.1103/physrevlett.125.085001] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2020] [Accepted: 07/20/2020] [Indexed: 06/11/2023]
Abstract
Warm dense matter (WDM)-an extreme state with high temperatures and densities that occurs, e.g., in astrophysical objects-constitutes one of the most active fields in plasma physics and materials science. These conditions can be realized in the lab by shock compression or laser excitation, and the most accurate experimental diagnostics is achieved with lasers and free electron lasers which is theoretically modeled using linear response theory. Here, we present first ab initio path integral Monte Carlo results for the nonlinear density response of correlated electrons in WDM and show that for many situations of experimental relevance nonlinear effects cannot be neglected.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-028262 Görlitz, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, D-01328 Dresden, Germany
| | - Michael Bonitz
- Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany
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Moldabekov ZA, Dornheim T, Bonitz M, Ramazanov TS. Ion energy-loss characteristics and friction in a free-electron gas at warm dense matter and nonideal dense plasma conditions. Phys Rev E 2020; 101:053203. [PMID: 32575188 DOI: 10.1103/physreve.101.053203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2020] [Accepted: 04/16/2020] [Indexed: 06/11/2023]
Abstract
We investigate the energy-loss characteristics of an ion in warm dense matter (WDM) and dense plasmas concentrating on the influence of electronic correlations. The basis for our analysis is a recently developed ab initio quantum Monte Carlo- (QMC) based machine learning representation of the static local field correction (LFC) [Dornheim et al., J. Chem. Phys. 151, 194104 (2019)JCPSA60021-960610.1063/1.5123013], which provides an accurate description of the dynamical density response function of the electron gas at the considered parameters. We focus on the polarization-induced stopping power due to free electrons, the friction function, and the straggling rate. In addition, we compute the friction coefficient which constitutes a key quantity for the adequate Langevin dynamics simulation of ions. Considering typical experimental WDM parameters with partially degenerate electrons, we find that the friction coefficient is of the order of γ/ω_{pi}=0.01, where ω_{pi} is the ionic plasma frequency. This analysis is performed by comparing QMC-based data to results from the random-phase approximation (RPA), the Mermin dielectric function, and the Singwi-Tosi-Land-Sjölander (STLS) approximation. It is revealed that the widely used relaxation time approximation (Mermin dielectric function) has severe limitations regarding the description of the energy loss of ions in a correlated partially degenerate electrons gas. Moreover, by comparing QMC-based data with the results obtained using STLS, we find that the ion energy-loss properties are not sensitive to the inaccuracy of the static local field correction (LFC) at large wave numbers, k/k_{F}>2 (with k_{F} being the Fermi wave number), but that a correct description of the static LFC at k/k_{F}≲1.5 is important.
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Affiliation(s)
- Zh A Moldabekov
- Institute for Experimental and Theoretical Physics, Al-Farabi Kazakh National University, 050040 Almaty, Kazakhstan
| | - T Dornheim
- Center for Advanced Systems Understanding (CASUS), Görlitz, Germany
| | - M Bonitz
- Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, 24098 Kiel, Germany
| | - T S Ramazanov
- Institute for Experimental and Theoretical Physics, Al-Farabi Kazakh National University, 050040 Almaty, Kazakhstan
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Dornheim T, Vorberger J, Groth S, Hoffmann N, Moldabekov ZA, Bonitz M. The static local field correction of the warm dense electron gas: An ab initio path integral Monte Carlo study and machine learning representation. J Chem Phys 2019; 151:194104. [DOI: 10.1063/1.5123013] [Citation(s) in RCA: 42] [Impact Index Per Article: 8.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022] Open
Affiliation(s)
- T. Dornheim
- Center for Advanced Systems Understanding (CASUS), Görlitz, Germany
- Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany
| | - J. Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, D-01328 Dresden, Germany
| | - S. Groth
- Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany
| | - N. Hoffmann
- Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, D-01328 Dresden, Germany
| | - Zh. A. Moldabekov
- Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany
- Institute for Experimental and Theoretical Physics, Al-Farabi Kazakh National University, Al-Farabi Str. 71, 050040 Almaty, Kazakhstan
| | - M. Bonitz
- Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany
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